Rational approximation to the fractional Laplacian operator in reaction-diffusion problems

This paper provides a new numerical strategy to solve fractional in space reaction-diffusion equations on bounded domains under homogeneous Dirichlet boundary conditions. Using the matrix transform method the fractional Laplacian operator is replaced by a matrix which, in general, is dense. The approach here presented is based on the approximation of this matrix by the product of two suitable banded matrices. This leads to a semi-linear initial value problem in which the matrices involved are sparse. Numerical results are presented to verify the effectiveness of the proposed solution strategy

A GCV based Arnoldi-Tikhonov regularization method

For the solution of linear discrete ill-posed problems, in this paper we consider the Arnoldi-Tikhonov method coupled with the Generalized Cross Validation for the computation of the regularization parameter at each iteration. We study the convergence behavior of the Arnoldi method and its properties for the approximation of the (generalized) singular values, under the hypothesis that Picard condition is satisfied. Numerical experiments on classical test problems and on image restoration are presented

Embedded techniques for choosing the parameter in Tikhonov regularization

This paper introduces a new strategy for setting the regularization parameter when solving large-scale discrete ill-posed linear problems by means of the Arnoldi-Tikhonov method. This new rule is essentially based on the discrepancy principle, although no initial knowledge of the norm of the error that affects the right-hand side is assumed; an increasingly more accurate approximation of this quantity is recovered during the Arnoldi algorithm. Some theoretical estimates are derived in order to motivate our approach. Many numerical experiments, performed on classical test problems as well as image deblurring are presented

Fast and accurate approximations to fractional powers of operators

In this paper we consider some rational approximations to the fractional powers of self-adjoint positive operators, arising from the Gauss-Laguerre rules. We derive practical error estimates that can be used to select a priori the number of Laguerre points necessary to achieve a given accuracy. We also present some numerical experiments to show the effectiveness of our approaches and the reliability of the estimates

Rational approximations to fractional powers of self-adjoint positive operators

We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in approximation theory involving Pad\ue9 approximants. The analysis improves some existing results and the numerical experiments proves its accuracy